Download Stochastic Analog Circuit Behavior Modeling by Point Estimation

Stochastic Analog Circuit Behavior Modeling by Point
Estimation Method
Fang Gong
University of California, Los
Angeles
Electrical Engineering
Department
Los Angeles, CA 90095, US
[email protected]
Hao Yu
Nanyang Technological
University
Electrical and Electronic
Engineering
[email protected]
ABSTRACT
Stochastic device parameter variations have dramatically
increased beyond the scale of 65nm and can significantly
lead to large mismatch for analog circuits. To estimate unknown analog circuit behavior in performance space under
the given stochastic variations in parameter space, many
state-of-art approaches have been developed recently. However, either Gaussian distribution or response surface model
(RSM) with analytical formulae has to be assumed when
connecting performance space and parameter space. A novel
point-estimation based approach has been proposed in this
paper to capture arbitrary stochastic distributions for analog circuit behaviors in performance space. First, to evaluate
high-order moments of circuit behavior in an accurate fashion, the point-estimation method has been applied with only
a few number of simulations. Then, probability density function (PDF) of circuit behavior can be efficiently extracted
by the obtained high-order moments. This method is further extended for multiple parameters under linear complexity. Extensive numerical experiments on a number of different circuits have demonstrated that the proposed pointestimation method can provide up to 181X runtime speedup
with the same accuracy, when compared with Monte Carlo
method. Moreover, it can further achieve up to 15X speedup
over the RSM-based method such as APEX with the similar
accuracy.
ACM Classification Keywords: B.7.2: - Integrated CircuitsDesign Aids
General Terms: Algorithms, Performance
Authors Keywords: Behavior Modeling, Point Estimation, Circuit simulation.
1.
INTRODUCTION
As semiconductor industry enters into nano-technology
node, large process variations become inevitable and hence
pose a serious threat to both analog circuit design and man-
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ISPD’11, March 27–30, 2011, Santa Barbara, California, USA.
Copyright 2011 ACM 978-1-4503-0550-1/11/03 ...$10.00.
Lei He
University of California, Los
Angeles
Electrical Engineering
Department
Los Angeles, CA 90095, US
[email protected]
ufacturing [1, 2, 3, 4]. Device variables in parameter space,
such as the effective channel length and threshold voltage
of transistors, can deviate significantly from nominal values
due to large uncertainties from chemical mechanical polishing (CMP), etching, lithography and etc. Under such circumstance, circuit behaviors in performance space can differ
from the nominal case by a large margin, which may further
lead to high loss of yield. With the process variation of
device variables in parameter space, it is desirable to extract the unknown distribution of variable circuit behaviors
in performance space. A robust circuit design and yield enhancement are especially important for analog circuits. One
critical but missing link here is how to find an efficient yet accurate mapping between parameter space and performance
space.
Note that the local random or stochastic variation is the
most difficult one to be calculated, which is also called mismatch for the behavior modeling of analog circuits. In the
past decade, many stochastic techniques had been proposed,
such as Monte Carlo simulation, linear regression [1], stochastic orthogonal polynomials (SoPs) expansion [5, 3], response
surface modeling based approaches [2, 4] and etc. The most
general approach is to apply the Monte Carlo (MC) simulation, which samples all variable parameters and then calculate stochastic analog circuit behaviors by a large number of
repeated simulations. As such, MC is too time-consuming to
be afforded for the post-layout verifications beyond 65nm.
To relieve the computational complexity, linear regression
method [1] has been deployed to approximate the circuit
behavior in performance space by a linear function of a number of normally distributed process variables in parameter
space, or Gaussian distribution. This approach is efficient
because of using analytical formula to obtain the circuit behavior. However, this approach cannot approximate nonnormal (non-Gaussian) distributions and might lead to the
loss of accuracy. With the use of different stochastic orthogonal polynomials (SoPs), SoP based methods can model process variations with non-Normally distributed random variables. The unknown distribution of circuit behaviors in performance space can be estimated by solving SoP expansion
coefficients [5, 3]. However, the SoP-based methods require
knowing the type of the stochastic distribution of the circuit
behavior. In practice, one known parameter distribution in
parameter space usually becomes unknown in performance
space after the mapping.
In order to capture unknown random distribution after
the mapping, response-surface-method (RSM) based methods [6, 4, 2] have been developed. One most important
work developed recently is asymptotic probability extraction
(APEX) [2] with the use of asymptotic waveform evaluation
[7]. This approach assumes a polynomial function of all process parameters and further applies moment matching to
extract the random distribution of circuit behavior (e.g. delay, gain, etc.). Nevertheless, the limitations of RSM-based
approaches can be summarized in two-fold. First, the circuit behavior in performance space has become one strongly
nonlinear function for random device variables. As such,
the extraction by RSM has become computationally expensive. Second, it is prohibitive to evaluate high-order moments E(f k ) with analytical formula of f , especially when
the number of random variables and the moment order k
increase. As such, the approaches based on RSM still cannot mitigate the super-linearly complexity while remaining
accuracy for large-scale problems.
In this paper, a new mapping algorithm is developed to
obtain the arbitrary circuit behavior in performance space
from the arbitrary device variable in parameter space. Highorder moments of circuit behavior f are first estimated by
Point Estimation (PE) method to efficiently characterize the
high-order moments E(f k ) by weighted-sum of a few sampled simulations. Therefore, one can significantly improve
the efficiency and accuracy of APEX [2] without the need to
assume RSM inputs. As a result, the distribution of circuit
behavior f in performance space can be efficiently obtained
by its moments E(f k ), calculated from the PE method in
the parameter space. Moreover, a normalized PDF function
is introduced so that to enhance the accuracy by eliminating
the potential round-off error. In addition, this approach has
been extended to consider the case with multiple parameters.
Extensive experiments on a number of different circuits
are performed to demonstrate the validity and efficiency of
our proposed algorithm. The contributions of this paper are
further clarified as follows. First, although point estimation
method has widely been applied for reliability analysis [8, 9],
it can only estimate at most four moments (e.g. the mean,
the variance, the skewness, the kurtosis) with empirical analytical formulae, and hence remains unclear how to estimate
moments with higher order. In this paper, a modified point
estimation method is developed to approximate higher order moments in a systematic manner. Moreover, unlike the
observed super-linearly complexity in RSM based methods,
our proposed method can be extended to deal with multiple
parameters with linear complexity, which is significant for
large-scale analog circuits.
The rest of this paper is organized as follows. In Section
2, we first review the mathematical formulation of the PDF
estimation and the moments used in response-surface-model
(RSM) methods. In Section 3, we introduce the point estimation (PE) method and further propose a new high-order
moments evaluation via PE. We also discuss one normalized PDF technique in Section 4 to reduce error and further
present experimental results in Section 5. This paper concludes in Section 6.
2.
BACKGROUND
2.1
Mathematical Formulation
We consider circuit behavior f with multiple random vari-
ables of process variations (x1 , x2 , · · · , xn ), which can be expressed as f (x1 , x2 , · · · , xn ). As such, parameter space can
be defined as the space Rn bounded by the min and max of
all random variables, and performance space R consists of
all possible behavior merits.
As a result of uncertainties in process technology, random
variables can deviate from their nominal values and lead
to variational circuit behavior. Our purpose is to extract
unknown distribution (e.g. PDF/CDF functions) of circuit
behavior by mapping the variable parameter distributions
in parameter space into performance space.
To this end, the probabilistic moments in both spaces
should be defined according to probability theory [10, 11]:
mpf = E(f p ) =
mpx = E(xp ) =
+∞
R
−∞
+∞
R
(f p · pdf (f ))df
(xp · pdf (x))dx
(1)
−∞
where mpf is the p-th moment of circuit behavior f in performance space, and mpx is the p-th moment of random variable
x in parameter space.
Lemma 1. Suppose pdf (f ) is continuous in performance
space. Then pdf (f ) can be determined uniquely by high order
moments E(f k ) (k = 1, 2, · · · , m).
Proof. Let Φ(ω) is the Fourier transform [12] of pdf (f )
and can be written as:
Φ(ω)
=
+∞
Z
³
´
pdf (f ) · e−jωf df
(2)
−∞
=
!
+∞Ã
Z
+∞
X
(−jωf )p
df
pdf (f ) ·
p!
p=0
−∞
=
+∞
Z
+∞
X
(−jω)p
·
(f p · pdf (f ))df.
p!
p=0
−∞
=
+∞
X
p=0
p
(−jω)
· mpf .
p!
As such, Φ(ω) can be expanded with high order moments
mpf , and pdf (f ) can be extracted from Inversion Fourier
transform of Φ(ω). In other words, there is a one-to-one correspondence between high order moments mpf and pdf (f ).
Notice that mpx in (1) can be computed accurately in parameter space with known pdf (x). Therefore, it is the key
problem to find an efficient mapping between mpx and mpf in
order to extract pdf (f ) in performance space.
2.2 Preliminary of PDF Calculation
The techniques to extract pdf (f ) with moments E(f k )
have been proposed in [2, 7], which will be reviewed in what
follows. First, time moments for f can be defined as:
(−1)k
·
mf =
k!
_k
+∞
Z
f k · pdf (f )df .
(3)
−∞
_k
It is clear that mf is defined in performance space ±but different from mkf in (1) due to a scaling factor (−1)k k!.
On the other hand, consider a linear time-invariant (LTI)
system H, and its time moments can also be defined as[7]:
_k
mt =
(−1)k
·
k!
+∞
Z
tk · h(t)dt.
(4)
−∞
where t is the time variable and h(t) is impulse response of
LTI system H. So, impulse response h(t) can be an optimal
approximation to pdf (f ) if we treat t as circuit behavior f
_k
_k
and make mt equal to mf . Furthermore, time moments in
(4) can be expressed as [7]:
_k
mt = −
M
X
ar
.
k+1
b
r=1 r
(5)
Where ar and br (r = 1, · · · , M ) are the residues and poles of
this LTI system, respectively. As such, the impulse response
of the LTI system can be simplified as:

M
k+1
 P
ar · ebr ·t (t ≥ 0)
h(t) =
(6)
 r=1
0
(t < 0)
In general, there are three steps to calculate h(t) as an
approximation to pdf (f ):
• Mapping mkx in parameter space into performance space
_k
as mf in (3).
_k
_k
• Make mf equal to mt and solve nonlinear equation
system in (5) for residues ar and poles br .
• Compute impulse response h(t) in (6) with residues ar
and poles br .
Clearly, one needs to find an efficient mapping between parameter space and performance space to obtain the stochastic circuit behavior. Within this mapping, the most challenging step is the evaluation of high-order moments. Although the RSM based methods, such as APEX, assume
that one nonlinear function can be found to approximate
the mapping, they might become unaffordable to calculate
the high-order moments for large-scale stochastic problems.
To this end, we have developed a modified point estimation
(PE) method to perform the mapping for the calculation of
high-order moments.
3.
HIGH ORDER MOMENTS ESTIMATION
In this section, we discuss how to evaluate high order moments of circuit behavior mkf by mapping parameter moments mkx from parameter space into performance space via
Point Estimation (PE) method.
3.1
Moments via Point Estimation
For illustration purpose, we consider circuit behavior f (x)
with single variable parameter x. Usually it is impractical to
compute mkf as (1) because pdf (f ) is unknown. The other
straightforward way is to use Taylor expansion, which involves high order derivatives. But there is no way to guarantee the existence of high order derivatives of f (x).
As such, we propose to leverage the Point Estimation
method to compute high order moments[8, 9], which approximates mkf with a weighted sum of sampling values of
f (x). Assume x̃j (j = 1, · · · , p) are estimating points of
random variable, and Pj are corresponding weights. In this
way, the k-th order moment of f (x) can be approximated
as:
mkf
+∞
Z
p
X
=
f k · pdf (f )df ≈
Pj · f (x̃j )k .
(7)
j=1
−∞
However, [8, 9] only provide empirical analytical formulae of
x̃j and Pj for first four moments. Therefore, it is significant
but remains unknown how to determine x̃j and Pj for higher
order moments systematically.
3.2 Estimating Points and Weights
To this end, we start with (7) in performance space, but
it is impossible to compute x̃j and Pj since both sides are
unknown. Thus, we need to reformulate the problem in parameter space, where random variable x and its distribution
(e.g. PDF function pdf (x)) are known beforehand.
According to classic probability theory[10, 11], we have
following theorem:
Theorem 1. Let x and f (x) are both continuous random
variables, and their
PDFs are pdf (x) and pdf (f ), respecR
tively. Suppose f k (x) · pdf (x)dx exists. Then
Z
Z
E(f k (x)) = f k (x) · pdf (f )df = f k (x) · pdf (x)dx
As such, the moments of circuit behavior f (x) in performance space can be calculated in parameter space. For
example, the k-th order moment of f (x) in equation (7) becomes:
Z
m
X
Pj · f (x̃j )k .
(8)
mkf = f (x)k · pdf (x)dx ≈
j=1
On the other hand, the k-th order moments of random
variable x can be written as:
Z
m
X
0
0
mkx = xk · pdf (x)dx ≈
Pj · (x̃j )k .
(9)
j=1
It is obvious that estimating points x̃j and corresponding
0
0
weights Pj in (8) are the same as x̃j and Pj in (9) because
they are all defined in parameter space. Therefore, we can
0
0
calculate x̃j and Pj from equation (9) with mkx obtained
from (1), and then estimate mkf using (8).
0
0
Now, the problem is how to solve for x̃j and Pj systematically. Since there are total 2m unknowns in (9), we need
to build 2m equations using first 2m moments of random
variable, which can be rewritten as:
m
P
j=1
m
P
j=1
m
P
j=1
···
m
P
j=1
0
Pj = 1 = m0x
0
0
Pj · x̃j = E(x) = m1x
0
0
0
0
Pj · (x̃j )2 = E(x2 ) = m2x
(10)
Pj · (x̃j )2m−1 = E(x2m−1 ) = m2m−1
x
Note that the right-hand-side of above nonlinear system are
first 2m moments of x in the behavior domain and can be
calculated exactly with known pdf (x) and definition in (1).
This nonlinear system (10) can be solved using algorithm
proposed in [7]. In what follows, we briefly describe this
algorithm.
0
0
Assume residues aj = Pj and poles bj = 1/x̃j , the equations (10) can be reformulated as:

 

a1 + a2 + · · · am
m0x
a1
a2
am


1
+
+
·
·
·

b1
b2
bm
mx

 
a1


 
2
+ ab22 + · · · ab2m

m
b2

=
x
m
1
2
(11)






.
.

  .
..



.
a1
a2
am
+ 2m−1
+ · · · 2m−1
m2m−1
x
b2m−1
b
b
of single variable xi with other variables set equal to mean
values. However, [8, 9] can only estimate first four moments
using explicit analytical formulae as linear combination.
Consider circuit behavior f (x1 , x2 , · · · , xn ) where x1 , x2 ,
· · · , xn are independent random variables, it is desirable to
estimate mkf (x1 ,x2 ,··· ,xn ) that is the k-th order moment of
f (x1 , x2 , · · · , xn ).
To estimate higher order moments of f (x1 , x2 , · · · , xn )
systematically, we derive the equation for mkf (x1 ,x2 ,··· ,xn ) as:
The system matrix of (11) is the well-known Vandermonde
matrix and can be divided into two parts:
Moreover, gi can be calculated as follows and detailed
derivation can be referred to technical report:
1
mkf (x1 ,x2 ,··· ,xn ) =
M · v = rhslow ; M · Λ−q · v = rhsupper
where rhslow consists of the low order moments (k = 0,1,
· · · , m − 1), and rhsupper contains the high order moments
(k = m, m + 1, · · · , 2m − 1). Λ−1 is a diagonal matrix of
{1/bj } (j = 1, · · · , m). And M matrix (m × m) can be
expressed as:


1
1
···
1
−1
−1
−1

b1
b2
···
bm−1 


−2
−2


b
b
·
·
·
b−2
1
2
m−1
M =

..
..
..
..




.
.
.
.
−(m−1)
b1
−(m−1)
b2
···
Therefore, the linear system M · v = rhslow can be solved
as v = M −1 · rhslow , and rhsupper = M · Λ−q · M −1 · rhslow .
Since M is also a Vandermonde matrix that is the modal
matrix for a system matrix in companion form, rhsupper
can be simplified as rhsupper = M̂ −q · rhslow , where M̂ −1 is


0 1 0 ···
0
 0 0 1 ···
0 


M̂ −1 =  .
(12)
.
.
.
.
..
..
..
..

 ..
.
s0 s1 s2 · · · sm−1
In this way, the eigenvalues of M̂ −1 are {1/bj } in (11). To
calculate M̂ −1 , we need to compute {st } (t = 0, · · · , m − 1)
with following equation system:


  m

mx
m0x
m1x · · · mm−1
s0
x
m+1 
2
m
 m1x




mx · · ·
mx

  s1

  mx
  ..

 =  ..
.
..
..
..
..

 .
  .

.
.
.
.
sm−1
m2m−1
mm−1
mm
· · · m2m−2
x
x
x
x
(13)
When {1/bj } are available, the {aj } can be calculated from
0
Equation (11). Therefore, the weights {Pj } and estimating
0
points {x̃j } can be computed systematically and used to
compute mkf in equation (8).
3.3
gi = c ·
Extension to multiple parameters
It is usually necessary to handle multiple variable parameters simultaneously in real-world problems. Thus, we discuss how to extend aforementioned techniques to deal with
multiple parameters.
Existing methods [8, 9] model mkf (x1 ,x2 ,··· ,xn ) as a linear
combination of moments mkf (xi ) , where f (xi ) is a function
(14)
∂ (f (xi ))
∂xi
For each parameter xi , we have its estimating points x̃j
and corresponding f (x̃j ). Hence, it is possible to calculate
∂ (f (xi ))/∂xi with finite difference method numerically.
Besides, the constant c can be computed with:
m0f (x1 ,x2 ,··· ,xn ) =
n
X
gi m0f (xi ) =
i=1
n
X
gi
(15)
i=1
,
n ∂ f (x )
∂ (f (xi ))
P
1
( i ).
=
c·
=1⇒c=
∂xi
∂x
i
i=1
i=1
n
X
−(m−1)
bm−1
gi mkf (xi ) .
i=1
m
2
n
X
As such, high order moments of multi-variable function
mkf (x1 ,x2 ,··· ,xn ) can be evaluated with moments of univariate function mkf (xi ) efficiently. Extensive experiments can
demonstrate its validity and efficiency.
3.4
Error Estimation
Theoretical maximum approximation error of point estimation method is analyzed in [13]. For the univariate case,
the maximum approximation error to exact integral value in
equation (1) can be governed by:
¯
¯
+∞
¯X
¯
Z
¯m
¯
k
k
¯
¯ ≤ α · k1/m . (16)
P
·
f
(x̃
)
−
f
(x)
·
pdf
(f
)df
j
j
¯
¯
¯ j=1
¯
−∞
where α is a constant, and k is the order of moments. m is
the number of estimating points x̃j for order k. As such, it
implies that more estimating points for each variable should
be used to reduce the estimation error of higher order moments.
4. PDF CALCULATION WITH MOMENTS
4.1 PDF/CDF Estimation with Moments
With high order moments mkf available, the next step is
to compute residues {ar } as well as poles {br } in (5) and
impulse response h(t) [2, 7] in (6). To do so, we calculate
_k
_k
mf in (3) and make them equal to mt in (5). The nonlinear
equation system becomes:





−



a1
b1
a1
b2
1
a1
b3
1
a1
b2M
1
+
+
+
aM
bM
aM
b2
M
aM
b3
M
a2
b2
a2
b2
2
a2
b3
2
+ ···
+ ···
+ ···
..
.
a2
+ b2M + · · ·
2
aM
b2M
M


PDF on all discrete points as:

_0
mf
_1
mf
_2
mf
M
P




.



 
 
 
 
=
  .
  .
  .
pdfnorm (fp ) =
(17)
M
P
_M
mf
Normalized PDF for Error Prevention
From (17), it is obvious that the accuracy of PDF approximation mainly depends on the accuracy of residues ar , poles
br and moments estimation. However, there are roundoff error within moment estimation and the PDF approximation,
which can lead to instability issue. In order to prevent potential error, we propose to normalize PDF calculated from
equation (6) to cancel out the potential roundoff error.
For illustration purpose, we take the roundoff error in moments as an example, and other roundoff error can be elim_k
inated with the same way. Assume mf is the exact value of
k-th time moment in equation (3), and m̃kf is the estimated
_k
value of k-th time moment. Also, we assume m̃kf = const·mf
due to roundoff error, where const is a scaling constant.
As such, when we use the direct solution in section 2,
the scaling constant in both system matrix and right-handside vector of (13) can be canceled out. Hence, st (t =
0, · · · , m − 1) and thus eigenvalues of equation (12) (that is
poles {1/bj }) are both exact values.
Next, the nonlinear equation system (17) becomes:
 _0 

 a1
M
+ ab22 + · · · abM
mf
b1
 _1 

 a21 + a22 + · · · a2M
 mf 


b1
b2
bM

 _2 
 a1




+ ab32 + · · · ab3M
b3
(18)
−
 = const ·  mf 
1
2
M




..

 ..


.
 .



a1
2
M
_M
+ ba2M
+ · · · ba2M
b2M
m
1
2
f
M
Which leads to ãj = const · aj , where aj are exact values of
residues. Therefore, PDF of f (x) is approximated with:
pdf (f ) =
M
X
_k+1
ãr · e b r
·f
r=1
=
M
X
_k+1
const · ar · e b r
·f
r=1
M
X
_k+1
const · ar · e b r
·fp
=
·fp
_k+1
const · ar · e b r
(21)
·fp
(20)
r=1
To normalize it, we can divide it with the total value of
_k+1
ar · e b r
r=1
N P
M
P
·fp
_k+1
ar · e b r
·fp
p=1 r=1
In this way, the scaling constant can be eliminated from
the approximation of PDF and thus normalization improves
the numerical stability of proposed algorithm.
4.3 Error Estimation
Since the Fourier transform in equation (3) is unique, it
is equivalent to evaluate the error of Φ(ω) in order to investigate the accuracy of PDF approximation with qth order
moments. It is ideally to compare the difference between
Fourier transform of estimated PDF and that of exact PDF.
However, the exact PDF is usually not available. Instead,
we use approximation with n+1 order moments as the exact
value, and proceed to estimate the error.
¯
¯ q+1
¯ Φ (ω) − Φq (ω) ¯
¯
(22)
Error = ¯¯
¯
Φq+1 (ω)
¯ q+1
¯
q
¯ P (−jω)p
¯
P
(−jω)p
¯
· mpf −
· mpf ¯¯
p!
p!
¯
p=0
¯ p=0
¯
= ¯
¯
q+1
¯
¯
P (−jω)p
p
¯
¯
·
m
f
p!
¯
¯
p=0
¯
¯
!
Ã
−1 ¯
¯
q+1
p
X
¯
¯ (−jω)q+1
mpf
(−jω)
¯.
= ¯¯
·
· q+1
¯
(q
+
1)!
p!
m
¯
¯
f
p=0
When |mpf | ≥ |mq+1
| ( p ≤ q + 1), above error estimation
f
can become:
¯
Ãq+1
!−1 ¯¯
¯
X (−jω)p
¯
¯ (−jω)q+1
¯.
Error ≤ ¯¯
·
(23)
¯
p!
¯ (q + 1)!
¯
p=0
The same error estimation can be obtained for |mpf | ≤ |mq+1
|
f
by taking reciprocal of circuit behavior to shift the behavioral distribution.
As such, the error estimation can be used to measure
the accuracy of the approximation with first q-th order moments. When the approximation order increases, the error
estimation should move to higher order as required.
4.4
(19)
In order to eliminate the scaling constant, we propose to
normalize PDF of f (x) as follows: First, we discretize f (x)
into discrete points {fp (x)} (p = 1, · · · , N ). As such, PDF
on p-th discrete point {fp (x)} can be expressed as:
pdf (fp ) =
r=1
N P
M
P
p=1 r=1
which has a Vandermonde matrix similar to (11) and can
be solved with the same technique in [7]. With poles {br }
and residues {ar } available, PDF function can be approximated with equation (6).
Notice that impulse response h(t) is zero for t < 0, but
the PDF in real-life problems can be nonzero for f ≤ 0. In
this case, PDF function can be shifted as [2] which can be
demonstrated with our experiments.
4.2
_k+1
const · ar · e b r
Complexity Analysis
The Monte Carlo simulation requires to generate massive
samples to cover the entire parameter space evenly, so number of simulations is pn where p is the number of samplings
for every single variable and n is the total number of random
variables.
As for RSM based methods, we take APEX as an example
where RSM formulation is the most time-consuming part.
For example, when RSM uses n random variables and k
order polynomial function, it has total Cn1 + Cn2 + · · · + Cnk
terms and thus APEX requires Cn1 +Cn2 +· · ·+Cnk simulation
samples. In other words, the complexity of APEX is O(nk ).
When proposed algorithm handles total n variables and
needs m estimating points for each variable, the total complexity is (m − 1) ∗ n + 1 (usually m ¿ n) or O(n). m − 1
denotes that estimating points of each variable includes the
nominal point and nominal circuit simulation can be shared
by all variables. Therefore, it has linear complexity.
0.9
5.
0.4
5.1
0.7
0.6
0.5
EXPERIMENTAL RESULTS
We have implemented proposed algorithm in the MATLAB environment, and all experiments are carried out on a
Linux server with a 2.4GHz Xeon processor and 4GB memory. We use a six-transistor SRAM cell and a two-stage operational amplifier to compare the accuracy and efficiency of
proposed algorithm with APEX [2] and Monte Carlo simulation. As an illustration, we consider the threshold voltages
of MOSFETs as independent random variables subject to
process variations, but our algorithm can also handle other
variation sources.
SRAM Cell
We first consider a typical design of 6T SRAM cell in
Fig.(1) and investigate the access time failure of the SRAM
cell during reading operation, which is determined by the
voltage difference between BL B and BL.
Monte Carlo with variations
Nominal analysis
0.8
Tmax
0.3
0.2
0.1
0
0
2
4
6
8
10
12
Figure 2: BL B discharge behavior with Vth variations
to deal with rare event in an SRAM. Therefore, we focus on
reducing average error of the performance distribution in the
entire range. We start from univariate case to validate proposed algorithm and then extend it to multi-variable case.
We have implemented three other methods for comparison:
• Monte Carlo simulation (MC): This is direct Monte
Carlo simulation.
WL=1
Vdd
• APEX: Implementation of asymptotic probability extraction algorithm proposed in [2].
+5V
• Point Estimation Method (PEM): Proposed algorithm that leverages the point estimation method.
Mn2
BL_B=1
Mp6
Mp5
Q =1
Q_B=0
Mn4
5.1.1 Univariate Case
BL=1
Mn3
Mn1
GND
First, we consider one random variable (e.g. threshold
voltage variation on M n2 ) to compare the accuracy of APEX
and PEM against MC. The random distributions (PDF function) from these methods are plotted in Fig.(3). Note that
the histogram from Monte Carlo simulation has been normalized to eliminate the effect of total number of samplings.
Figure 1: Schematic of SRAM 6-T Cell
0.16
Initially, both BL B and BL are pre-charged to V dd,
while Q B stores zero and Q stores one. When reading the
SRAM cell, BL B starts to discharge from V dd and produces a voltage difference ∆V between it and BL. The time
it takes BL B to produce a large enough voltage difference
is called access time. Since process variations are inevitable,
the access time of manufactured SRAM cells can deviate
from nominal value. When access time is larger than acceptable maximum value Tmax , this leads to an access time
failure.
In our experiment, we consider threshold voltages of all
MOSFETs as independent variables which are normally distributed with 30% perturbation from nominal values. As
such, there are perturbations to the nominal discharge trajectory on BL B as shown in Fig.(2). Therefore, we can
investigate the access time failure by capturing the random
distribution of voltage on BL B at Tmax time-step: when
voltage of BL B at Tmax is larger than its nominal value,
the access time failure happens.
Note that both PEM and APEX can not capture high
precision in the tail region of CDF/PDF, which is required
0.14
PEM
APEX
0.12
0.1
0.08
0.06
0.04
0.02
0
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
Figure 3: random distributions of BL B voltage at
Tmax (Monte Carlo result has been normalized)
When compared with Monte Carlo results, PEM can provide better accuracy than APEX, especially in the peak
and right tail regions. However, APEX has better efficiency
even if the same order of moments are used: APEX needs
only 8.59 second with quadratic function in response surface model, while PEM requires 17.71 seconds with seven
estimating points for one variable.
It is because APEX use analytical formula which is suitable for low dimensions. It should be noticed that the number of required simulation samples in APEX will increases
exponentially when more variables or a strongly nonlinear
RSM required.
5.1.2
Multiple Variable Parameters
Next, we consider all threshold voltages of six transistors
are independent random variables, which are normally distributed with 30% perturbation from nominal values. Similarly, we attempt to compare all three methods, but it is
prohibitive to implement APEX as response surface model
becomes very complicated, especially when high order response surface model is required.
Instead of original APEX, we calculate high order moments numerically using results from Monte Carlo simulations and extract PDF with technique in APEX, which is
denoted as MMC+APEX. The random distributions from
all methods are compared in Fig.(4).
10 independent variables are used to model random variation for each transistor [14]. As such, RSM using quadratic
function has 1830 coefficients and thus APEX requires 1830
simulation samples as discussed in Section 4.4. However,
PEM only needs 121 simulation samples when 3 estimating
points are used for each independent variable, and achieves
15X speedup over APEX.
5.2 Operational Amplifier
We further consider a two-stage operational amplifier in
Fig. (5) and a negative feedback circuit in Fig.(6). We use
this example to show that proposed method can estimate
random distributions where circuit behavior is negative.
Vdd
+5V
Mp5
Mp8
Mp7
Output
Mp1
Input-
Mp2
Input+
Is
Mn6
Mn4
Mn3
Vss
0.14
-5V
Figure 5: Schematic of Operational Amplifier
0.12
PEM
MMC+APEX
0.1
Similar to SRAM cell example, we consider threshold voltages of all MOSFETs as independent variables which are
normally distributed with 30% perturbation from nominal
values. Note that the threshold voltages of input transistor pair (M p1 , M p2 ) should be kept the same to ensure the
convergence of nonlinear system solver in circuit simulators.
0.08
0.06
0.04
R2
0.02
R1
0
0.08
0.1
0.12
0.14
0.16
0.18
Output
0.2
Input
Figure 4: random distributions of BL B voltage at
Tmax (Monte Carlo result has been normalized)
It is obvious that PEM can capture the exact distribution
of BL B voltage at Tmax , which fits well with Monte Carlo
simulation and MMC+APEX method. This can shows that
PEM can achieve very high accuracy in the multi-variable
problems. Also, we compare only the runtime of MC and
PEM in Table 1, since runtime of MMC+APEX is almost
the same as MC. In this table, PEM uses 5 estimating points
for each variable parameter, and can achieve the same accuracy with 119.2X speedup over Monte Carlo method.
Table 1: Runtime Comparison of three methods
Method
3
Monte Carlo (3 × 10 )
PEM (5 point)
Time (second)
Speedup
7644
64.12
1x
119.2x
To compare the efficiency with APEX, we can consider a
SRAM cell under commercial 65nm CMOS process where
Figure 6: Schematic of a unity gain feedback circuit
There are a number of op am specifications in time-domain
and frequency-domain, such as slew rate, settling time, phase
margin, input offset voltage and etc. In our experiment, we
investigate the input offset voltage variation due to threshold voltage variations of MOSFETs.
In this experiment, we implement following methods for
comparison purpose:
• Monte Carlo simulation (MC): This is direct Monte
Carlo simulation.
• Moments from Monte Carlo (MMC+APEX):
Estimate high order moments of input offset voltage
from Monte Carlo simulation, and extract the PDF
using techniques in [2, 7].
• Point Estimation Method (PEM): Proposed algorithm that leverages the point estimation method.
First, we validate the accuracy of PEM by comparing the
random distributions of input offset voltage from different
methods in Fig.(7). PEM employs 5 estimating points for
each variable and achieve the correct PDF function by shifting distribution of circuit behavior into positive region and
moving it back.
0.35
MMC+APEX
PEM
0.3
7. REFERENCES
0.25
0.2
0.15
0.1
0.05
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Figure 7: random distributions of input offset voltage (Monte Carlo result has been normalized.)
Since MMC+APEX method has the exact moment values from Monte Carlo simulations, it can provide very high
accuracy. Also, PEM can offer the same accuracy with
MMC+APEX method, which implies that PEM can achieve
high accuracy of high order moments. On the other hand, we
further validate accuracy of PDF function from PEM against
the histogram from Monte Carlo simulation in Fig.(7), which
fit with each other very well.
Moreover, we compare the runtime of Monte Carlo method
and PEM in Table (2), and MMC+APEX is omitted because
it has almost the same runtime as Monte Carlo method.
Besides, we list PEM method with different number of estimating points for each variable to demonstrate it has linear
scalability with the estimating points at the same time.
Table 2: Runtime Comparison between Monte Carlo
simulation and PEM
Method
3
Monte Carlo (3 × 10 )
PEM (3 point)
PEM (9 point)
Time (second)
Speedup
27765 (7.71 hours)
153.1 (0.04 hours)
525.84 (0.15 hours)
1x
181.5x
52.8x
From Table (2), PEM can provide up to hundreds times
speedup over MC method. Moreover, the computational
cost increases linearly with the number of estimating points
for each variable.
6.
ically reduce the computational cost seen in APEX. Moreover, the proposed method can be extended to deal with
multiple parameters under linear complexity. Experiments
on a few different circuits have shown that the proposed
method can provide up to 181X more runtime speedup with
the same accuracy when compared with the Monte Carlo
method. Also, it can achieve up to 15X speedup over the
RSM based method such as APEX with the similar accuracy.
CONCLUSION
In this paper, we have proposed one efficient point estimation (PE) based algorithm to extract the stochastic circuit
behavior in performance space from parameter space. Our
approach can perform an efficient evaluation of high-order
moments of circuit behavior, and thus circumvent the use of
response surface model (RSM) methods. This can dramat-
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